Beta (aka simple linear regression) has long been a widely used measure of single name and portfolio exposure to a benchmark. One of the drawbacks of beta, however, is that it lumps in all returns, regardless of direction, into the same analytical basket. Those wishing to measure index exposure isolating negative portfolio return observations are mostly out of luck. The best they can do is pick a historical time period where the asset or portfolio of interest lost money most of the time.

What is needed, therefore, is a version of beta that only includes observations where an asset exclusively experiences negative returns. The objective is to understand what the asset's exposure to a benchmark is purely on days where it loses value. In portfolio manager-speak, this translates into: "On days when we lost money, what was our exposure to the market?". This is an interesting data point for many alpha-seeking stock pickers: their objective is to find single name stocks that enjoy gains when the market is bid and yet do not wholly participate in market crashes.

The Mechanics of Filtered Beta:

Although we have been talking about a "downside" measurement, in truth, the generalized case involved here is the filtering of return observations according to arbitrary criteria. For example, we may choose to look at every return in a stock above 0.5%. Therefore, downside beta is merely a special case where the returns are selected according to the following rule:

Xi < 0

Were each Xi is a return observation in the target asset's return history. For example, if an index and a portfolio have the following returns:

Index

Portfolio

+0.004

-0.006

+0.010

+0.007

+0.018

+0.007

+0.004

+0.007

+0.002

-0.010

+0.001

-0.003

-0.005

+0.028

For downside beta, only the returns highlighted in bold will be used in the resulting beta calculation. These are the set of returns that correspond to days when the portfolio lost money. Let's take a look at a real-world example using downside beta:

Citigroup (NYSE:C) Vs. S&P500 Index

Taking daily data for both Citi and the S&P since Jan 1, 2013 results in a standard beta of 1.60. Assuming this beta is sound, this tells us that on a given day, we can expect Citi's returns to be about 60% more volatile than the index's returns. If the index has 1% loss, Citi's stock value should decrease by 1.6%. Since we are looking at ALL returns (both positive and negative), we expect this statement to hold true for up days in the market: i.e. if the market rallies 1%, Citi should rally even more: 1.6%.

Now let's look at downside beta for the same period. Applying the filtering mechanism above, we find that the downside beta is down to 1.15. This is good news for a portfolio manager who is long Citigroup. What we've learned here is that on days where the stock loses value, its returns were very nearly in line with those of the market. In other words, when we lost money in our Citi position, in percentage terms, our losses did not exceed the market's losses.

The filtered beta feature is currently only available in RiskAPI Enterprise and will soon be released in the RiskAPI Add-In

Of note are the negative returns (shown in blue in the chart above) since May 1, with the largest being a jaw-dropping -7.32% day on May 23rd.

Volatility shown is rolling, 90-day realized. Calculations are as of 6/13/2015, executed on daily data since 12/31/2012, in JPY. Volatility as calculated is the annualized standard deviation of lognormal daily returns.